We use the classical Brezzi-Douglas-Marini (BDM) interpolator
\(\Pi _{BDM}\)[1]} to project \(\hbox{$u$}\)  and
\(\hbox{$\eta $}^s\)  onto the finite element spaces \(\hbox{$V$}^k_{h,0}\)  and
\(\hbox{$V$}^{k,s}_{h,0}\) . We denote \(\Pi _Q\)  as the \(L^2\) -projection onto the finite
element space \(Q_h^{k-1}\) . Note that due to the commuting projection property, we have:
\((\nabla \cdot \Pi _{BDM}\hbox{$\eta ^s$},q_h^s)_s&=(\nabla \cdot \hbox{$\eta ^s$},q_h^s)_s, \quad &&\forall q_h^s\in Q_h^{k-1,s},\\\nabla \cdot \Pi _{BDM}\hbox{$u$}^f|_{\Omega ^f}&=\Pi _{Q}(\nabla \cdot \hbox{$u$}^f)|_{\Omega ^f}=0.\) 
